By George Chrystal

This Elibron Classics booklet is a facsimile reprint of a 1904 version through Adam and Charles Black, London.

**Read Online or Download Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 PDF**

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**Additional resources for Algebra: An Elementary Text-Book for the Higher Classes of Secondary Schools and for Colleges. Part 1 **

**Example text**

S(X,o) = S(X). S(Xn) < (1) with equality if Taking X. = 0 n S is additive. 1, we have S(Xi,Xi-1) §5. THE MORSE INEQUALITIES Let M be a compact manifold and 29 a differentiable function f on M with isolated, non-degenerate, critical points. < ak Let critical points, and i Mak M. = Then e% 1 Mai-1) = H (Mai-1 H*(Mat Mat-1) where is the index of the critical point, X. ), . H*(e 1,e 1) by excision, coefficient group in dimension otherwise. C M ak = M with k Xi S = R. we have R),(Ma1,Ma1-1) = CX; i=1 where C.

We will use the following theorem, which we will not prove. is §6. 1 (Sard). If M1 and M2 are differentiable manifolds having a countable basis, of the same dimension, then the image of the M2 is of class C1, and f: M1 set of critical points has measure A critical point of in 0 M2- is a point where the Jacobian of f f is For a proof see de Rham, "Variotes Differentiables," Hermann, singular. Paris, 1955, p. 10. 2. PROOF: E: N -s Rn. is M. N We have just seen that is a focal point iff x x the point For almost all x e Rn, not a focal point of is an n-manifold.

This completes the PART II A RAPID COURSE IN RIEMANNIAN GEOMETRY §8. Covariant Differentiation The object of Part II will be to give a rapid outline of some basic concepts of Riemannian geometry which will be needed later. For more infor- mation the reader should consult Nomizu, "Lie groups and differential geoMath. Soc. Japan, 1956; Helgason, "Differential geometry and sym- metry. metric spaces," Academic Press, 1962; Sternberg, "Lectures on differential geometry," Prentice-Hall, 1964; or Laugwitz, "Differential and Riemannian geometry," Academic Press, 1965.